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Math Help - difficult sin x function

  1. #1
    Senior Member pacman's Avatar
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    difficult sin x function

    Solve for x: (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27
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  2. #2
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    Quote Originally Posted by pacman View Post
    Solve for x: (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x} = 26/27
    Something wrong here? If you put y = \sin^2x then 0\leqslant y\leqslant1. But the function (16/81)^y + (8/9)^{1-y} is greater than 1 throughout the interval [0,1], so it can never be equal to 26/27.
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  3. #3
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    ok say instead of 26/27 it was 3/2.

    how can you solve it for an exact soln? w/o using any approximating methods
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  4. #4
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    Quote Originally Posted by Krahl View Post
    ok say instead of 26/27 it was 3/2.

    how can you solve it for an exact soln? w/o using any approximating methods

    If It is  ( \frac{4}{9})^{\cos^2{x}} not

     ( \frac{8}{9})^{\cos^2{x}}

    Sub (\frac{4}{9})^{\sin^2{x}} = u  then

     <br />
u^2 + (4/9) u^{-1} = M<br />

     u^3 - M u + (4/9) = 0

    It becomes a cubic equation

    after solving u  u = (\frac{4}{9})^{\sin^2{x}}

     x = \arcsin{ \sqrt{ \frac{ \ln{u} }{ \ln(4/9) } } }
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  5. #5
    Senior Member pacman's Avatar
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    i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?
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  6. #6
    Senior Member pacman's Avatar
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    and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?
    Attached Thumbnails Attached Thumbnails difficult sin x function-sin-x-2.gif  
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  7. #7
    Super Member Failure's Avatar
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    Quote Originally Posted by pacman View Post
    i plot this one: y = (16/81)^(sin^2 x) + (8/9)^(cos^2 x) for x =0 to pi radian. In my graph it looks like this, any comments?
    My comment: it is not true that this is the plot of y = (16/81)^{\sin^2 x} + (8/9)^{\cos^2 x}. Rather, this is the plot of y = (16/81)^{sin^2 x} + (8/9)^{cos^2 x}-26/27.

    Also: This function assumes a minimum value of \tfrac{19}{18}\approx 0.235 at x=\tfrac{\pi}{2}+n\cdot\pi, n\in\mathbb{Z}. But it is always >0.
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  8. #8
    Super Member Failure's Avatar
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    Quote Originally Posted by pacman View Post
    and this one, (16/81)^(sin^2 x) + (8/9)^(cos^2 x) = 26/27; what does it mean?
    It means that this time you have plotted y=(16/81)^{sin^2 x} + (8/9)^{cos^2 x} for a larger range of x and, for some reason that I cannot fathom, seem to have surprised yourself with the result.
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  9. #9
    Senior Member pacman's Avatar
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    your observation is quite alright with me, thanks
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